Why do we have trolley/buggy bans during the winter?

Buggies are most likely to be banned when the course is very wet. This is largely for health and safety reasons, but also to protect the areas where buggies are likely to cause excessive wear and rutting.
Trolleys and single seat buggies (and therefore larger 2 seat buggies) are most likely to be banned during periods of heavy frost. This is done to protect the grass by reducing wear on it.